In A Geometric Sequence, S ∞ = 3 2 S_{\infty}=\frac{3}{2} S ∞ ​ = 2 3 ​ And S 3 = 14 9 S_3=\frac{14}{9} S 3 ​ = 9 14 ​ . Calculate The First Three Terms Of The Sequence.

$$$$

Introduction

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of an infinite geometric sequence can be calculated using the formula $S_{\infty} = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. In this problem, we are given that the sum of the infinite geometric sequence is $\frac{3}{2}$ and the sum of the first three terms is $\frac{14}{9}$. We need to find the first three terms of the sequence.

Formula for the Sum of an Infinite Geometric Sequence

The formula for the sum of an infinite geometric sequence is given by:

$$S_{\infty} = \frac{a}{1 - r}$$

where $a$ is the first term and $r$ is the common ratio.

Formula for the Sum of the First n Terms of a Geometric Sequence

The formula for the sum of the first n terms of a geometric sequence is given by:

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Given Information

We are given that the sum of the infinite geometric sequence is $\frac{3}{2}$ and the sum of the first three terms is $\frac{14}{9}$. We can use this information to set up two equations:

$$\frac{3}{2} = \frac{a}{1 - r}$$

$$\frac{14}{9} = \frac{a(1 - r^3)}{1 - r}$$

Solving for the Common Ratio

We can start by solving the first equation for $a$:

$$a = \frac{3}{2}(1 - r)$$

Substituting this expression for $a$ into the second equation, we get:

$$\frac{14}{9} = \frac{\frac{3}{2}(1 - r)(1 - r^3)}{1 - r}$$

Simplifying this equation, we get:

$$\frac{14}{9} = \frac{3}{2}(1 - r^3)$$

Multiplying both sides by $\frac{2}{3}$, we get:

$$\frac{28}{27} = 1 - r^3$$

Subtracting $1$ from both sides, we get:

$$-\frac{1}{27} = -r^3$$

Dividing both sides by $-1$, we get:

$$\frac{1}{27} = r^3$$

Taking the cube root of both sides, we get:

$$r = \frac{1}{3}$$

Solving for the First Term

Now that we have found the common ratio, we can substitute it into one of the original equations to solve for the first term. We will use the first equation:

$$\frac{3}{2} = \frac{a}{1 - r}$$

Substituting $r = \frac{1}{3}$, we get:

$$\frac{3}{2} = \frac{a}{1 - \frac{1}{3}}$$

Simplifying this equation, we get:

$$\frac{3}{2} = \frac{a}{\frac{2}{3}}$$

Multiplying both sides by $\frac{2}{3}$, we get:

$$\frac{3}{1} = a$$

So, the first term is $a = 3$.

Finding the Second and Third Terms

Now that we have found the first term and the common ratio, we can find the second and third terms. The second term is found by multiplying the first term by the common ratio:

$$a_2 = ar = 3 \times \frac{1}{3} = 1$$

The third term is found by multiplying the second term by the common ratio:

$$a_3 = ar^2 = 1 \times \frac{1}{3} = \frac{1}{3}$$

Conclusion

In this problem, we were given that the sum of the infinite geometric sequence is $\frac{3}{2}$ and the sum of the first three terms is $\frac{14}{9}$. We used this information to find the first three terms of the sequence. The first term is $a = 3$, the second term is $a_2 = 1$, and the third term is $a_3 = \frac{1}{3}$.

Introduction

Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will answer some frequently asked questions about geometric sequences, providing a deeper understanding of this concept.

Q: What is a Geometric Sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the Formula for the Sum of an Infinite Geometric Sequence?

A: The formula for the sum of an infinite geometric sequence is given by:

$$S_{\infty} = \frac{a}{1 - r}$$

where $a$ is the first term and $r$ is the common ratio.

Q: What is the Formula for the Sum of the First n Terms of a Geometric Sequence?

A: The formula for the sum of the first n terms of a geometric sequence is given by:

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How Do I Find the Common Ratio of a Geometric Sequence?

A: To find the common ratio of a geometric sequence, you can use the formula:

$$r = \frac{a_{n+1}}{a_n}$$

where $a_n$ is the nth term and $a_{n+1}$ is the (n+1)th term.

Q: How Do I Find the First Term of a Geometric Sequence?

A: To find the first term of a geometric sequence, you can use the formula:

$$a = \frac{S_n}{1 - r^n}$$

where $S_n$ is the sum of the first n terms, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the Difference Between a Geometric Sequence and an Arithmetic Sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic sequence, on the other hand, is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

Q: How Do I Determine if a Sequence is Geometric or Arithmetic?

A: To determine if a sequence is geometric or arithmetic, you can look at the ratio between consecutive terms. If the ratio is constant, the sequence is geometric. If the difference between consecutive terms is constant, the sequence is arithmetic.

Q: What are Some Real-World Applications of Geometric Sequences?

A: Geometric sequences have many real-world applications, including:

• Compound interest: Geometric sequences can be used to calculate compound interest on investments.
• Population growth: Geometric sequences can be used to model population growth.
• Music: Geometric sequences can be used to create musical patterns and rhythms.
• Art: Geometric sequences can be used to create geometric patterns and designs.

Conclusion

In this article, we have answered some frequently asked questions about geometric sequences, providing a deeper understanding of this concept. We have covered topics such as the formula for the sum of an infinite geometric sequence, the formula for the sum of the first n terms of a geometric sequence, and the difference between a geometric sequence and an arithmetic sequence. We have also discussed some real-world applications of geometric sequences.